Polar decomposition, singular-value decomposition, and Autonne-Takagi factorization Carlton M. Caves1, ∗ 1Center for Quantum Information and Control, University of New Mexico, Albuquerque NM 87131-0001, USA (Dated: February 14, 2017) This document provides an introduction to the polar decomposition, the singular-value decom- position, and Autonne-Takagi factorization of symmetric matrices, with an emphasis on how all of these are close to the same thing once one thinks about operators instead of their representations. We begin by reviewing the polar decomposition and the singular-value decomposition, since Autonne-Takagi fac- torization is a special case of these. p Polar decomposition. Given a linear operator M, the polar decomposition starts with the positive operators M yM p y and MM . These two positive operators have the same (real, nonnegative) eigenvalues, denoted as λj. The y y 2 p y eigenvalues of M M and MM are the squares, λj . If we let jeji and jfji be (orthonormal) eigenvectors of M M p y and MM , respectively, with jeji and jfji having the same eigenvalue λj, we have p y X M M = λjjejihejj , (1) j p y X MM = λjjfjihfjj . (2) j y y 2 y Now we notice that MM (Mjeji) = M(M Mjeji) = λj Mjeji, so Mjeji is an eigenvector of MM with eigenvalue 2 y λj . Thus, for eigenvectors jeji in the support of M M, i.e., for which λj 6= 0, we define jfji ≡ Mjeji/λj; this definition imposes a natural and unique way of pairing up eigenvectors jeji and jfji in degenerate subspaces and of phasing y all the eigenvectors jfji. For eigenvectors jeji in the null subspace of M M, we can start with any orthonormal eigenvectors in the null subspace of M yM and pair them up with any choice of orthonormal eigenvectors in the null subspace of MM y. With these choices, we can write X Mjeji = λjjfji () M = λjjfjihejj , (3) j y y X M jfji = λjjfji () M = λjjejihfjj ; (4) j The eigenvalues λj are called the singular values of M, and the eigenvectors jeji and jfji are called the right and left singular vectors. Letting U be the unitary operator that transforms between the two eigenbases, i.e., X Ujeji = jfji () U = jfjihejj , (5) j we have that U transforms between the two positive operators, i.e., p p U M yMU y = MM y . (6) This leads us to the polar decomposition, p p M = U M yM = MM yU , (7) p p M y = M yMU y = U y MM y . (8) ∗Electronic address: [email protected] 2 y 2 The unitary operator U is unique if M M is invertible (i.e., all the eigenvalues λj are nonzero). This is clear from the construction of the polar decomposition; moreover, when M yM is invertible, we have U = M(M yM)−1=2. p y p 2 For a Hermitian operator H, with eigenvalues hj and eigenvectors jeji, we have H H = H = jHj, i.e., λj = jhjj; the only role of the unitary U in the polar decomposition is to insert a sign change for negative eigenvalues of H: X U = sign (hj)jejihejj . (9) j A unitary operator U is its own polar decomposition. A normal operator N, i.e., N yN = NN y, has an eigendecom- P iφj position N = j λje jejihejj, where the λjs are nonnegative, so p y p y X N N = NN = λjjejihejj ; (10) j the role of the unitary U in the polar decomposition is to put the phases back in: X iφj U = e jejihejj . (11) j p What distinguishes normal operators is that the unitary U and the positive operator N yN in the polar decomposition commute. Singular-value decomposition. The singular-value decomposition is really the same thing as the polar decomposition, the only difference being that there is a standard basis jji in which we want the singular values to appear in diagonal form as X Λ = λjjjihjj . (12) j By introducing a unitary operator V that maps from the standard basis to the eigenbasis jeji, i.e., V jji = jeji, we can write p y X y M M = λjjejihejj = V ΛV (13) j and thus put the polar decomposition in the form M = W ΛV y , (14) where W = UV , with W jji = jfji. Equation (14) is the singular-value decomposition. Notice that in this construction, we have the freedom to rephase the right singular vectors jeji or, equivalently, to multiply V on the right by a unitary that is diagonal in the standard basis. In the standard basis, the singular-value decomposition becomes X y X Mjk = hjjMjki = hjjW jliλlhljV jki = hjjfliλlheljki , (15) l l with hjjW jli = hjjfli and hkjV jli = hkjeli being the unitary matrices that transform from the standard basis to the singular vectors. The singular-value decomposition is usually stated directly in terms of matrices, i.e., the representa- tions of operators in a standard basis, and this statement is that any matrix can be diagonalized by a pair of unitary matrices. Autonne-Takagi factorization. Autonne-Takagi factorization is usually stated in terms of a complex symmetric matrix; if we wish to state it in terms of operators, we need a standard basis jji relative to which transposition and complex conjugation are defined. Now let's state precisely what we want to prove: If M = M T is a symmetric operator, relative to a standard basis jji, there exists a unitary operator V such that T X V MV = Λ = λjjjihjj , (16) j 3 where the diagonal elements are the (real, nonnegative) singular values of M. Since M = V ∗ΛV y, this is a special case of the singular-value decomposition with W = V ∗. Translated to the polar decomposition, the unitary operator in the polar decomposition becomes U = WV y = V ∗V y = (VV T )y. Proof. Let S be a unitary operator such that SyM yMS = Λ2. Consider the symmetric operator L = ST MS = LT . One has LyL = SyM yS∗ST MS = SyM yMS = Λ2 and LLy = ST MSSyM yS∗ = ST MM yS∗ = (SyM yMS)∗ = Λ2. This implies that L is diagonal in the standard basis: X iφj X iφj X iφj =2 X iφj =2 L = λje jjihjj = e jjihjj Λ = e jjihjj Λ e jjihjj . (17) j j j j The second form is the polar decomposition of the normal operator L. The last form is the one we now use by writing X −iφ =2 T X −iφ =2 Λ = e j jjihjj S MS e j jjihjj . (18) j j Defining X −iφ =2 V = S e j jjihjj (19) j completes the proof. Notice that to define V , what we used was the above-mentioned freedom to rephase the right singular vectors, so that V also diagonalizes MM y. Indeed, using X ∗ X ∗ V = jejihjj , V = jej ihjj , j j (20) ∗ y X ∗ U = V V = jej ihejj , j we find that ∗ y p y X ∗ M = V ΛV = U M M = λjjej ihejj . (21) j Thus one may think of the real content of Autonne-Takagi factorization as the fact that the left and right singular ∗ vectors of a symmetric operator can be rephased so that they are complex conjugates of one another, i.e., jfji = jej i..